M.: Location of resonances generated by degenerate potential barrier, Electron

We study resonances of the semi-classical Schrodinger operator H = h 2 �+V on L 2 (IR N ). We consider the case where the potential V have an absolute degenerate maximum. Then we prove that H has resonances with energies E = V0 + e i � �+1h 2� �+1kj + O(h 2�+1 �+1 ), where kj is in the spectrum of some quartic oscillator.


1
Introduction and main results.
We are interested in this paper to study the asymptotic behavior for Schrödinger operator: on L 2 (IR N ), where V is a bounded real function having an absolute maximum V 0 realized at a unique point that we suppose to be x = 0.Under some assumptions on V, one can define on L 2 (IR N ) the operator H θ (x, hD) with domain the Sobolev space W 2 (IR N ) obtained by analytic dilation : Here U θ denotes the group of unitary operators on L 2 (IR N ), given by U θ u(x) = e − iNθ 2 u(e iθ x), for θ in IR such that 0 ≤ θ < θ 0 < π 2 .D denotes the differential operator (−i In [2] and [3] Ph.Briet, J.M.Combes and P.Duclos consider the same problem with V having a non degenerate maximum at x = 0.They prove existence of resonance values of H near 0. They prove too that these values are located outside a ball of radius O(h).In the second paper they give the first factor in the asymptotic expansion of resonance values of H.We are interested here to generalize results of [2] and [3] to cases of degenerate potential barrier of order σ i.e V (x) ∼ −(λ Here σ > 1 is an integer and λ 1 , . . .λ N some strictly positive real constants.We prove that resonance values of H are located outside a ball centered at 0 and of radius O(h 2σ σ+1 ).This power of h exists already in the results of Martinez-Rouleux [5].The authors study there asymptotic of eigenvalues of the operator H in the case of degenerate minimum of order σ for V .They prove that H has no eigenvalue inside a ball centered at 0 and of radius O(h 2σ σ+1 ).In fact this result is closely related to the Taylor expansion of V near 0. Our second aim is to get more precise result under more precise assumption on the Taylor expansion of V .We notice that in a recent paper [1] we study the resonances of H in the one dimensional case where the potential V have a degenerate maximum of quartic type.We give in that paper the full asymptotic expansion of resonance values near the barrier maximum.The method used there is the BKW techniques.This method is specific to the one dimensional case.Here we use local analysis on the resolvent of the operator.To state the results assume the following hypothesis on V : (A 1 ) V is a bounded real function having an absolute degenerate maximum of order σ at x = 0. (A 2 ) V θ (x) = V (e θ x), θ ∈ IR has an analytic continuation to complex θ in S α = {θ ∈ I C, | Imθ |< α} as a family of bounded operators.
(A 4 ) Near x = 0 , V θ has the Taylor expansion : ) for all θ ∈ S α , with λ 1 , ...λ N non vanishing positive real constants.Consider the operator given by : It is well known that the spectrum of K is discrete.We use (k j ) j≥1 for eigenvalues of K. We recall that the operator H θ has a domain D(H θ ) = W 2 (IR N ) independent of θ.This operator has an analytic continuation to an analytic family of type A in S α (see [4] and [6], section XIII).We get : Théorème 1 Consider the operator H θ 0 = −h 2 e −2θ 0 ∆+V θ 0 and assume the hypothesis ) such that for h small enough , H θ 0 has purely discrete spectrum inside B(h) with total algebraic multiplicity equal to the multiplicity of k j .
In the direction to get the full asymptotic expansion of resonance values of H, we try more subtle approach and we get the following result: Assume (A 5 ) Near x = 0 , V θ has the Taylor expansion : with p > 0 an integer.
) such that for h small enough, H θ 0 has purely discrete spectrum inside B(h) with the same algebraic multiplicity as k j .
Finally to interpret the eigenvalues of H θ 0 mentioned in the Theorem 1 as resonance values of H we state the following theorem : Théorème 3 Let φ be in the domain D of analytic dilation dense in L 2 (IR N ).For all C > 0 and h small enough ((H − z) −1 φ, φ) has meromorphic continuation from I C + = {z ∈ I C, Imz > 0} into a complex disk centered at 0 with radius Ch 2σ σ+1 .The poles of this continuation belong to the set of the eigenvalues of H θ 0 given in theorem 1.
To prove theorems we decompose the operator H into a direct sum of an interior operator and an exterior one.To do this we use j i and j e ∈ C ∞ (IR N ), with j i = 0 outside the ball B(0, h 1 σ+1 δ i ) et j e = 0 inside the ball B(0, h 1 σ+1 δ e ), satisfying to j 2 i + j 2 e = 1.Here 0 < δ e < δ i are two positive constants to be chosen later.The support of ∇j i and ∇j e is located in support of j e ), we define the application J by We have JJ * = id H .For Imθ = 0, we define where on L 2 (IR N ) and support of j e ) with Dirichlet conditions on the boundary of the support of j e .Hence we have J(D(H d θ )) ⊂ D(H θ ) and we can write the resolvent equation: where Π is the operator given by Π = H θ J − JH d θ acting on H d as follows: where w = (V θ + e 2σθ (λ For the study of the interior operator H i θ one prove the following lemma: ii) The spectrum of H i θ is θ independent, purely discrete and given by : Proof : The proof of i) can be reduced to prove the same properties for the one dimensional operator : ).To prove that the family (h θ ) is analytic of type A, one proves that h θ is closed for all θ, 0 < Im(θ) < π σ+1 .For this we have to establish for all θ = iβ such that 0 <| β − π 2σ+2 |< π 2σ+2 , the following inequality : on D(h θ ), where c is a positive constant.We have: We write One can estimate the second integral in this equation as follows : By integration by parts we get: Let B(R) denotes the ball centered at 0 with radius R > 0 and B(R) c the complementary set of this ball.We write : EJQTDE, 2006 No. 18, p. 5 This yields in combination with (6): For the first integral in the equation 5 : We now write : Therefore by ( 7) and (8)we get : Finally choosing R big enough we get (4).
To prove ii), notice that H i θ has a compact resolvent for 0 < Imθ < π σ+1 .By analyticity of (H i θ ) θ the spectrum of ).Using the scaling of order h 1 σ+1 we get: In the exterior domain our operator H here has the same shape as this one studied in [3].Here we localize more closely to the boss of the barrier.So we get more precise results than lemma II-5 in [3].
Proof: We first notice that by assumption A 4 there exists > 0 independent of h such that for θ 0 = iβ 0 with β 0 > 0 small enough we get for all | x |< : Therefore we get ∀x such that h for a convenient C > 0 independent of h.Hence by this and assumption A 3 we get: Let u ∈ D(H e θ 0 ) be such that u = 1, and let z ∈ Υ.Since (∆u, u) has a real value, we get by (9): , where (., .)denotes the inner product in L 2 (IR N ).Hence we get: This proves that Ker(H e θ 0 − z) = {0} and (H e θ 0 − z) has a closed image.On the other hand (H e θ 0 − z) * = H e θ0 − z, and we get by the same way: ( Therefore Ker(H e θ 0 −z) * = {0}, and hence the image of (H e θ 0 −z) is the whole L 2 (supportj e ).This leads to i).For ii) it is immediately given by (10).EJQTDE, 2006 No. 18, p. 7 This lemma means that the operator valued function z → (H e θ 0 − z) −1 is holomorphic inside Υ.By the same argument one gets the following: The operator valued functions z → g(H d θ 0 − z) −1 J * and z → (H θ 0 − z) −1 g have no poles inside Υ for a convenient C.
Let now k j ∈ σ(K) and γ denotes the closed curve given by: Here E d = e −i π σ+1 h 2σ σ+1 k j and ρ > 0 is a constant independent of h to be chosen later small enough.We have: Lemme 4 Let γ be the closed curve defined by (11), then for ρ small enough we have by assumptions (A 1,2,3,4 ) : i) γ is in the resolvent set of By our localization formula close to the boss of the barrier, the proof of this lemma seems to be much more complicated than if we do this using standard localization formula as in [3].This is largely due to the contribution of the commutator part in Π in the equation (3).The estimation of this contribution becomes here not precise.By the same way as in [3] we get the same results.To avoid needless repetition we shall not rewrite proof.

Proof of Theorems
The following lemma shows that the spectrum of H θ 0 is discrete inside γ.The total algebraic multiplicity is equal to the multiplicity of E d as an eigenvalue of H i θ 0 .Lemme 5 Let γ be the closed curve given by (11), with ρ small enough but independent of h.Then under assumptions (A 1,2,3,4 ) for h small enough, P θ 0 and P i θ 0 have the same rank.where P θ 0 and P i θ 0 denote the projectors defined respectively by: uniformly for z ∈ γ.We have: By lemma 1 ii) one can choose ρ such that dist(γ, σ(H i θ 0 )) > ch ) on the support of j i .Then we get: This equation in combination with lemma 4 ii) leads to (12).Since | γ |= O(h 2σ σ+1 ), we get : σ+1 ).
This leads to : and then smaller than 1 for h small enough, and the lemma is proved.
EJQTDE, 2006 No. 18, p. 9 To finish the proof of theorem 1, let E be an eigenvalue of H θ 0 inside γ, and let φ be the corresponding normalized eigenvector.We have : By equation ( 3) and the fact that the commutator part has no contribution to the integral we get: Then Therefore equation (12) leads to This proves the theorem 1.To prove theorem 2, Assuming moreover (A 5 ), the equation ( 14 ).This in combination with the theorem 1 proves the theorem 2.